Poincaré Group Spin Networks

Spin network technique is usually generalized to relativistic case by changing SO \(\varvec{(4)}\) $\mathbf{(}\mathbf{4}\mathbf{)}$ group – Euclidean counterpart of the Lorentz group – to its universal spin covering SU \(\varvec{(2)}\times \) $\mathbf{(}\mathbf{2}\mathbf{)}\times$ SU \(\varvec{(2)}\) $\mathbf{(}\mathbf{2}\mathbf{)}$ , or by using the representations of SO \(\varvec{(3,1)}\) $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{1}\mathbf{)}$ Lorentz group. We extend this approach by using inhomogeneous Lorentz group \(\varvec{\mathcal {P}}= {\varvec{SO}}\varvec{(3,1)}\rtimes \mathbb {R}^4\) $\mathcal{P}=\mathit{SO}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{1}\mathbf{)}⋊{\mathbb{R}}^4$ , which results in the simplification of the spin network technique. The labels on the network graph corresponding to the subgroup of translations \(\mathbb {R}^4\) ${\mathbb{R}}^4$ make the intertwiners into the products of SU \(\varvec{(2)}\) $\mathbf{(}\mathbf{2}\mathbf{)}$ parts and the energy-momentum conservation delta functions. This maps relativistic spin networks to usual Feynman diagrams for the matter fields.